The difference between Gaffe and Mistake

When used as nouns, gaffe means a foolish and embarrassing error, especially one made in public, whereas mistake means an error.

Mistake is also verb with the meaning: to understand wrongly, taking one thing or person for another.

check bellow for the other definitions of Gaffe and Mistake

Gaffe as a noun:

A foolish and embarrassing error, especially one made in public.

Mistake as a noun:

An error; a blunder.
Mistake as a noun (baseball):

A pitch which was intended to be pitched in a hard-to-hit location, but instead ends up in an easy-to-hit place.

Mistake as a verb (transitive):

To understand wrongly, taking one thing or person for another.

Examples:

"Sorry, I mistook you for my brother. You look very similar."
Mistake as a verb (transitive, obsolete):

To misunderstand (someone).
Mistake as a verb (intransitive, obsolete):

To commit an unintentional error; to do or think something wrong.
Mistake as a verb (obsolete, rare):

To take or choose wrongly.